What the removed roof buys

Fukinuki yatai — the blown-off roof — is the convention in Japanese narrative painting of drawing a building with its roof and often its near wall simply absent, so that the rooms are open to the viewer from above and to one side. It is usually explained as a device for showing what is happening inside, which is true and is a statement about narrative rather than about geometry.

The geometric question is different and has a number attached. A pinhole camera can also be placed so as to see into a roofless building. What can it not do that the convention does?

The answer turns out not to be see more, which was the first hypothesis and was wrong in an instructive way. It is see evenly — and evenness is a property no projection through a centre has, at any distance, for any arrangement of rooms.

The first answer was wrong, and the way it was wrong is the finding #

The obvious hypothesis is that the parallel system sees more. It does not, necessarily, and the search that was set up to prove it did something more interesting.

The first version of this measurement swept a grid of eye positions — five heights, four radii, twelve bearings — and took the one that saw the most floor. The assertion was that the parallel system would beat all of them. It failed, and it failed because the search walked to the far corner of its own grid: coverage rises with the eye’s distance, monotonically, because a distant eye is a nearly parallel one.

The search did not find a rival to the convention. It found the convention. That is a result rather than a failed assertion, and it reframes the whole comparison: the pinhole and the parallel system are not two alternatives, they are two ends of one family, and the parallel system is the limit of the pinhole as the eye recedes. Asking which sees more is asking where on a continuum to stand.

So the honest comparison is not against the best eye in an unbounded search. It is against the eye that would actually make the picture — one placed so that the building fills the frame — and the quantity that separates them is not how much is seen but how evenly.

What a wall hides, and why the answer has no position in it #

Under a parallel system every sightline runs in the same direction, so the strip of floor a wall hides is

$$w = \frac{h}{\tan\theta}$$

with $h$ the wall’s height and $\theta$ the elevation of the view. There is no distance in that expression. A wall at the near end of the building and a wall two hundred metres along hide strips of exactly the same width, because the sightline that clears one clears the other at the same angle.

That is the property, and it is what makes the convention work on a building of any length — a narrative scroll running through a mansion can show every room on the same terms, and a reader can compare the third room with the tenth without correcting for anything.

Under a pinhole, the same strip is $h \cdot d / (H - h)$ with $d$ the distance from wall to eye and $H$ the eye’s height, so it depends on where in the building the wall stands. The near rooms and the far rooms are shown on different terms and there is nothing in the picture to say by how much.

Two measurements, and both are exact on one side #

The comparison produces two numbers and each one has a zero on the parallel side, which is what makes the pinhole’s number readable.

Coverage spread across rooms. The parallel system reaches 55.6% of every room’s floor and the spread between the best-served room and the worst is exactly zero — not small, zero, because the samples are the same samples at the same relative positions and the occlusion test gives the same answer. The framing eye reaches 49.4%, 55.6% and 49.4%, a spread of 6.2 percentage points.

Area spread across the floor. A square metre of floor images to the same area everywhere under the parallel system: the ratio of the largest imaged square metre to the smallest is 1.000000000000. Under the framing eye it is 1.535. So the same floor tile is drawn half as large again in one room as in another, and a reader with a ruler measuring the far room gets an answer 53% wrong.

The first number is about what can be seen and the second about what can be measured, and the convention wins both by having no dependence on position at all.

Why this is a fact about the building rather than the picture #

There is a subtlety worth drawing out, because it explains why the convention appears where it does.

Both numbers above are zero-on-one-side because a parallel projection is translation invariant: moving an object sideways moves its image sideways and changes nothing else. A pinhole is not, and the departure grows with how much of the scene is off the optical axis.

A building of one room is nearly on axis and the two systems very nearly agree. A building of three rooms is the figure above. A palace of twenty rooms, or a narrative that runs along a corridor for several metres of paper, is a case where the pinhole’s numbers vary by a factor rather than by a few points — and it is exactly the case the convention is used for.

So fukinuki yatai is not a solution to the problem of seeing inside one room. It is a solution to the problem of showing many rooms on comparable terms, which is what a narrative handscroll of a mansion needs and what no single station point provides. That is the same shape of finding as the handscroll’s — a convention that looks like a limitation turns out to be the answer to a problem perspective cannot solve — and the two conventions come from the same milieu and the same objects.

The elevation is the only knob, and it is a real one #

The parallel system has one free parameter that affects coverage — the elevation of the view — and it is worth working through, because it is where the convention’s choices actually live.

At 34° the strip a wall hides is 3.4 m and a 4.2 m room shows under a fifth of its floor. At 52° the strip is 1.8 m and the room shows 55.6%. At 80° it is 0.4 m and the room shows 90%. Coverage rises monotonically with elevation and reaches 100% only when the view is straight down, at which point the standing figures collapse to nothing and the picture stops being about people.

Coverage has a closed form and it is worth writing down, because it says where the knob’s ends are. A room of depth $L$ behind a wall of height $h$, viewed at elevation $\theta$, shows

$$1 - \frac{h}{L\,\tan\theta}$$

of its floor. On the figure’s building — $h = 2.3$ m, $L = 4.2$ — that gives 18.8% at 34°, 57.2% at 52°, 75.6% at 66° and 90.3% at 80°, against the walk’s own 55.6%, 77.8% and 90%. Two routes, agreeing to the grid’s own resolution, which is the same pairing the strip formula and the sightline test make one section above.

Two ends fall out of it.

There is a hard floor to the elevation. Coverage reaches zero when $\tan\theta = h/L$, which here is $\arctan(2.3/4.2) = 28.7°$ — below that a wall hides the whole room and the convention shows nothing at all. That is not a gradual fading; it is a definite angle, computable from the building’s own proportions before any picture is drawn.

And the returns fall off fast. From 34° to 52° buys thirty-eight points of coverage; the next fourteen degrees buy eighteen; the fourteen after that, fifteen. The curve rises steeply out of its cutoff and flattens, so the useful range of elevations is narrow and its lower end is sharp.

The expression also says that the only thing about the building that enters is the ratio $h/L$ — how tall the walls are against how deep the rooms are. A hall with low partitions is generous and a corridor of tall screens is not, and the elevations these paintings actually use are therefore readable as a statement about the architecture rather than about the painter. That is the same kind of inference the removed roof’s own measurement makes from the other side, where a reader recovers a room’s proportions from the page.

So there is a trade inside the convention as well as between it and perspective, and it is a trade between how much floor is visible and how much of a standing figure survives. The angles these pictures actually use — steep enough to see in, shallow enough to draw faces — are the compromise, and it is a compromise with two computable ends.

What the elevation does not affect is uniformity. At every angle in the sweep, the spread across rooms is exactly zero. The knob buys coverage and cannot buy or lose the property the convention exists for, which is a clean separation and is the reason the two are measured separately.

The near wall, counted #

The closing section notes that removing the near wall as well takes coverage to 100%, and the arithmetic behind that is worth one paragraph because it says how much of the convention’s own liberty is doing the work.

With the roof off and the near wall standing, the floor a reader sees is $1 - h/(L\tan\theta)$ — 55.6% at the figure’s elevation, so the wall is costing 44 points. Remove it and every sightline reaches the floor unobstructed, so coverage is exactly one at every elevation above zero, and the cutoff at 28.7° disappears with it.

So the two editing operations are not equal. Removing the roof buys the ability to look in at all; removing the near wall buys the last 44%, and it buys it uniformly rather than as a function of the angle. A painter who has removed both has a picture whose coverage no longer depends on the elevation, which frees the elevation to be chosen entirely for how the figures read — which is presumably why the angles in these paintings cluster where they do rather than at the steep end the coverage curve would recommend.

It also sharpens what the convention is not. A single removed roof leaves a picture whose readability depends on the architecture, through $h/L$; a removed roof and a removed wall leave one that does not. The second is a plan with elevations standing on it, and it is the arrangement a carpet drawn with its people upright reaches by a different route and the field’s own comparison table puts in the same row.

What the convention costs #

Every system in this field buys something and pays for it, and this one’s bill is worth stating plainly.

It gives up the station point. There is no distance from which a fukinuki yatai picture is a correct projection of the building, because a parallel projection is the limit of a projection whose centre has gone to infinity, and infinity is not a place in the room. The figures in this field print no viewing distance for the same reason the parallel field’s do not.

It gives up the depth cue that size provides. A room at the far end of the building is drawn the same size as the near one, so nothing in the picture says which is further. The next essay in this field is about what has to carry depth instead.

And it gives up the near wall as well as the roof. The measurement above removes only the roofs and still loses 44% of every floor to the walls, because a wall on the viewer’s side of a room hides the strip behind it whatever system is used. Real fukinuki yatai pictures often remove the near wall too, and that is not a further liberty — it is the same operation applied to the second occluder, and it takes the coverage to 100% by removing the last thing in the way.

That third point is worth dwelling on, because it says what the convention actually is. It is not a projection with a special property. It is an ordinary parallel projection plus an editing operation — delete the surfaces between the viewer and the subject — and the reason it is worth measuring is that the editing operation is only available to a system whose geometry survives it. Delete the roof from a perspective picture and the rooms are still shown on different terms; delete it from a parallel one and the picture is a plan with elevations attached, which is a measurable object.

The same operation in a photograph, and why it is not available #

It is worth asking what the equivalent operation would be for a camera, because the answer explains why the convention belongs to the parallel systems and not to perspective.

A photographer wanting to show three rooms at once has three options and each fails differently.

Remove the roof and photograph from above. This works and produces a picture in which the rooms are shown on different terms — the 6.2-point spread and the 1.535× area ratio measured here. The picture is a projection, is evidence, and is not comparable across rooms.

Photograph each room separately and lay the pictures out. Now every room is shown well and there is no single geometry relating them; the layout is a convention. This is aspective, applied to rooms rather than to body parts, and it is what an estate agent’s floor plan does.

Use a very long lens from very far away. This approaches the parallel system in the limit, uniformity and all — and it requires a viewpoint at a distance and a height that generally do not exist, and a lens that compresses the picture until the walls hide everything, because the strip a wall hides grows as the elevation falls and a distant eye is a low one unless it is also very high.

The third option is the interesting one, because it says the convention is not unobtainable by a camera — it is the limit of one. What makes it a drawing system rather than a photographic technique is that a draughtsman can simply be at infinity, and a photographer cannot.

The measurement’s own limits #

Three, and the second is the one that would change the numbers most.

The walls are solid and full height. Real rooms of this kind are divided by screens, sliding panels and open bays, and a real fukinuki yatai painting shows some of those and removes others. The 55.6% above is for a hard case; a building of open bays would report more, under both systems, and the spread — which is the finding — would be unchanged, because it comes from the projection rather than from the walls.

The elevation is a free parameter and the coverage depends on it. At 34° a parallel system reaches a third of each room and at 80° four fifths. The figures state the elevation they use and the drag runs it. What does not depend on it is the zero.

And the occlusion test is a sightline walk over a sample grid, so its resolution is one row of samples — which is why the strip formula is computed independently and required to agree. The agreement is to within one row across the whole elevation range, which is what a sampling error looks like when it is the only error present.

The site’s own gate carries one more check on the same machinery, and it is the refusal: a building with no walls at all must hide nothing. It reports 100.0% reached. Without that, a coverage number would be consistent with an occlusion test that was quietly rejecting samples for some reason unrelated to walls — and a test that never returns everything is visible is a test that has not been shown to be measuring visibility.

The two numbers, once more #

The comparison worth keeping is with what isometric actually means. A cutaway buys visibility by removing matter; an axonometric buys it by choosing a direction no face is parallel to. Both are answers to the same question, and only the first admits that something was taken away.